By P.R. Halmos

A variety of the mathematical writings of Paul R. Halmos (1916 - 2006) is gifted in Volumes. quantity I involves study courses plus papers of a extra expository nature on Hilbert area. the rest expository articles and the entire renowned writings look during this moment quantity. It contains 27 articles, written among 1949 and 1981, and likewise a transcript of an interview.

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An example of an ergodic T(z) ~ = az on K, where a T w h i c h is not w e a k - m i x i n g is not a root of unity. ) (iv) There are examples of w e a k - m i x i n g mixing. T which are not strong- Kakutani has an example c o n s t r u c t e d by c o m b i n a t o r i a l methods, and M a r u y a m a c o n s t r u c t e d an example using Gaussian processes. and K a t o k - S t e p i n also have examples. bility space, let ~(X) if (X,B,m) is a proba- denote the c o l l e c t i o n of all invertible measure-preserving transformations ~(X) Indeed, Chacon with the "weak" t o p o l o g y of (X,8,m).

By p o s i t i v i t y , and hence Therefore UFN(X) + f(x) a max l~n~N f (x) n = max 0ANON f (x) n when FN(X) = FN(X). Thus We have f ~ F N - UF N on A = {x: FN(X) > 0}, so > 0 for 33 IA f d m >- IA FNdm - IA UFNdm = ;X FNdm - ;A UFNdm since FN : 0 >- ;X FNdm - IX UFN dm since FN t 0 = >_ 0 since HUH -< i. on X\A. UF N e 0. 5 hold if U = UT for measure- T. 6: T: X ~ X be measure-preserving. 1 g E LR(m) If i n-i sup ~ [ g(Tm(x)) n~l m=0 B e = {x EX: and > e} then I gdm ~ em(B flA) B NA if and T-IA : A and a m(A) < ®.

N i=0 iff for all space and f (L2(m) 1 iff for all set becomes Then n-i I n i=0 T becomes is a probability ! 9: Suppose (a) of time. independent on the average. The next result Theorem asymptotically, is strong-mixing f,g for all (L2(m), f (L2(m), iff (U~f,g) + (f,l)(1,g) (U~f,f) ~ (f,l)(l,f). 0 is 43 Proof: Ca), Cb), and (c) are proved using similar methods. shall prove (c) to illustrate the ideas. proof will prove (2) = (i). (i) = (3). (XA,I)(I,XB). This follows by putting f =XA, g =XB, for A,B E B.

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